(\partial)/(\partial x)(x^{x^2-y^2})

解

\frac{\partial}{\partial x} (x^{x^{2} - y^{2}})

x^{x^{2} - y^{2}} (2 x ln (x) + \frac{x ^{2} - y ^{2}}{x})

解答ステップ

\frac{\partial}{\partial x} (x^{x^{2} - y^{2}})

y

を定数として扱う

指数の規則を適用する:

a^{b} = e^{b l n (a)}

x^{x^{2} - y^{2}} = e^{(x^{2} - y^{2}) l n (x)}

= \frac{\partial}{\partial x} (e^{(x^{2} - y^{2}) l n (x)})

連鎖法則を適用:

e^{(x^{2} - y^{2}) l n (x)} \frac{\partial}{\partial x} ((x^{2} - y^{2}) ln (x))

= e^{(x^{2} - y^{2}) l n (x)} \frac{\partial}{\partial x} ((x^{2} - y^{2}) ln (x))

\frac{\partial}{\partial x} ((x^{2} - y^{2}) ln (x)) = 2 x ln (x) + \frac{x ^{2} - y ^{2}}{x}

= e^{(x^{2} - y^{2}) l n (x)} (2 x ln (x) + \frac{x ^{2} - y ^{2}}{x})

e^{(x^{2} - y^{2}) l n (x)} = x^{x^{2} - y^{2}}

= x^{x^{2} - y^{2}} (2 x ln (x) + \frac{x ^{2} - y ^{2}}{x})